Part of the Atlas of Small Regular Polytopes

Polytope of Type {3,6,3}

Atlas Canonical Name {3,6,3}*324a

Overview

Group
SmallGroup(324,39)
Rank
4
Schläfli Type
{3,6,3}
Vertices, edges, …
3, 27, 27, 9
Order of s0s1s2s3
9
Order of s0s1s2s3s2s1
6
Also known as
7T4(1,1)(3,0). if this polytope has another name.

Special Properties

  • Universal
  • Locally Toroidal
  • Orientable
  • Flat

Quotients maximal quotients in bold

3-fold

9-fold

Covers minimal covers in bold

2-fold

3-fold

4-fold

5-fold

6-fold

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

P/N, where N=<(s1*s2)^2> of order 3

5 facets

3 vertex figures

  • 3 of 3-fold non-regular quotient of {6,3}*108

Representations

Permutation Representation (GAP) 
s0 := (2,3)(5,6)(8,9);;
s1 := (2,3)(5,6)(7,8);;
s2 := (4,7)(5,8)(6,9);;
s3 := (1,4)(2,5)(3,6);;
poly := Group([s0,s1,s2,s3]);;
Finitely Presented Group Representation (GAP) 
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(9)!(2,3)(5,6)(8,9);
s1 := Sym(9)!(2,3)(5,6)(7,8);
s2 := Sym(9)!(4,7)(5,8)(6,9);
s3 := Sym(9)!(1,4)(2,5)(3,6);
poly := sub<Sym(9)|s0,s1,s2,s3>;
Finitely Presented Group Representation (Magma) 
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s1*s0*s1*s0*s1, s2*s3*s2*s3*s2*s3, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1 >;

References

  1. Theorem 11E7, McMullen P., Schulte, E.; Abstract Regular Polytopes (Cambridge University Press, 2002)

to this polytope.